Plane to circle get "pinched" corners
I personally don't see any other solution than delete the inner ellipse and re-extrude inwards, bevel, etc.
If you want to redo this kind of shape from scratch, here are several ways to do it, the less destructive would be with a curve, but if you begin with a mesh you can now custom the profile:

Create your ellipse object, select the top edge:



Bevel it with CtrlB, give it enough segments:



Enable the Custom option and draw your profile:



A similar custom profile has also been implemented for the curves, go in the Properties panel &gt; Object Data &gt; Geometry &gt; Bevel and click on Profile:

You will have much more control on the bevel shape if you use the Object option though:

Or of course you can use a mesh as a profile along a curve with the Array and Curve modifiers: